definitions of "linear" (with application to Sig Figures)

[John Denker <jsd@MONMOUTH.COM>]

At 10:51 AM 2/1/00 -0500, Richard Bowman wrote in part:
>
> 4. The usual "rules" only apply to linear functions.

I would like to remind everybody that there exist two grossly inconsistent
definitions of "linear".

Definition 1:  A "linear transformation" (A) obeys the rules
        A(x + y) = A(x) + A(y)
and (for any scalar t)
        A(t x) = t A(x)

Definition 2:  A "linear equation" is anything of the form
        A(x) = m x + b

--------

In particular note a linear equation represents a linear transformation if
and ONLY if the y-intercept (b) happens to be zero.

Constructive suggestions:  To avoid ambiguity:
  *) A linear equation can be called a first-order polynomial, or
(preferably) a first-degree polynomial.  It can also be called an affine
transformation.
  *) A linear transformation can be called a proportionality relationship.

-----------

Also at 10:51 AM 2/1/00 -0500, Richard Bowman wrote in part:
>
>  I let my students simply
> treat trig functions as if they are linear. Thus three digits in angle
> implies three digits in sin or cos, etc.

That version of the rule applies only to proportionality laws.  It doesn't
even apply to first-degree polynomials.  It certainly isn't a good way to
think about trig functions.  Example:  suppose we know the angle
        theta = 89.3(1)                 [roughly 3 sig digs]
then we can calculate
        tan(theta) = 82(13)             [roughly 1 sig dig]
        sin(theta) = 0.999925(23)       [roughly 5 sig digs]

This example does _not_ depend on nonlinearity.  Nonlinearity is rarely
more than a minor contribution to the error propagation.  A more relevant
thing to consider is the magnitude of the first derivative [dA/dx] and its
relationship to the chordal slope [A(x)/x].  Tan(theta) is very steep
compared to its chordal slope, while sin(theta) is very nearly horizontal
(given the aforementioned value of theta).